CAREER: Conformal Symmetry in Theoretical Physics

Yale University, New Haven CT

Investigators

Abstract

This CAREER award is for an integrated program of research, education, and outreach focusing on quantum field theories with conformal symmetry (CFTs). The primary goals are: 1) To further our knowledge of how conformal symmetry constrains correlation functions in CFTs. While the forms of 2-point and 3-point functions are fixed by conformal symmetry, 4-point functions are not completely determined, and can in general be expanded into a basis of functions called conformal blocks. Knowing how to perform these expansions explicitly is extremely important for formulating basic consistency conditions (called crossing symmetry) in any field theory with conformal symmetry. The PI will develop the theory of conformal blocks, focusing on situations (e.g., correlation functions of operators with spin and CFTs in odd dimensions) where these decompositions are poorly understood. 2) To perform detailed analytical and numerical studies of the implications of crossing symmetry and unitarity (the conformal bootstrap) in CFTs. These studies will produce general rigorous bounds on physical observables (operator dimensions) and develop techniques to predict their values in a wide variety of physically relevant theories. By pursuing these studies, the PI is seeking to map out the space of CFTs, identify new CFTs and learn about known ones, make quantitative predictions for experimentally realizable condensed matter systems, explore the validity of ideas in physics beyond the standard model, and test and formulate new theoretical conjectures related to the AdS/CFT correspondence and quantum gravity. Conformal invariance is a generalization of scale invariance that only requires that the angles between corresponding curves do not change. Conformal field theories (CFT) lie at the heart of theoretical physics, linking all the fields of physics represented within the Physics Division as well as other programs in other Divisions in NSF. They are ubiquitous in the space of quantum field theories; they holographically describe (connect properties in a volume to the information encoded in the surface of that volume) consistent theories of quantum gravity; they are realized in a wide variety of real-world condensed matter systems such as the critical three-dimensional Ising model of ferromagnetism in terms of the coupling between spins of neighboring atoms; and they may even play an important role in physics beyond the Standard Model, the concept that accurately describes the known elementary particles in terms of quarks, leptons and gauge fields. There are wide classes of physically relevant CFTs where conventional field theory techniques such as perturbation theory and large-N expansions cannot be used. The recursive conformal bootstrap approach is currently underdeveloped and appears to be an extremely promising way of making quantitative predictions for physical observables in these otherwise intractable theories. Broader Impacts : The education and outreach component of this program will introduce students at all education levels to the power of symmetry in physics. 1) The PI is developing a series of hands-on demonstrations illustrating principles of symmetry in physics, being made available to middle and high school students in racially diverse and high poverty school districts through Yale's "Math Mornings" lecture series and during a "Theoretical Physics Day" hosted by the PI. 2) The PI is developing a public lecture on "Symmetries in Physics" to be given in multiple venues. 3) The PI is developing and teaching two distinct 3-day Modules for Yale's Pathways Science Collaborative Hands-On Learning and Research (SCHOLAR) program, one focusing on symmetries of space and time and one focusing on exploring scale invariance and extra dimensions. 4) The PI is actively involving Yale undergraduates in his research as well as including their participation in outreach activities. 5) The PI is developing a new graduate level course on "Conformal Field Theories", introducing Yale students to modern research topics, giving them opportunities to develop their oral presentation skills, and producing a set of lecture notes that will be disseminated publicly.